What is the “kissing problem” that has plagued mathematicians for centuries?

It all started in the 16th century with the famous explorer or pirate (depending on your point of view) Sir Walter Raleigh. You might be surprised to read the title, because he’s neither a mathematician nor, as far as we know, a problem with kissing.

All he had were cannon balls, and one question: What was the best way to stack them to minimize the space they took up on their ships.

It’s a math problem, and in math those bullets are spheres and “kisses” are called points where one sphere touches another.

Raleigh’s question would create a mathematical mystery that would occupy brilliant minds for hundreds of years.

On a trip to America in 1585, the great mathematician Thomas Harriot asked the scientific advisor who had given him the solution:

The best way to store your cannonballs is to arrange them in a pyramid shape.

In a 1591 manuscript, Harriet created a table for him that calculated the number of cannonballs to be placed at the base of a pyramid with a triangular, square, or elliptical base.

But Harriet kept thinking about the subject, so she considered the implications of the atomic theory of matter then in vogue.

Commenting on the theory in his friendly correspondence with the famous astronomer Johannes Kepler, he noted the problem of packing.

Kepler thought that the optimal way to minimize the space left by the spaces between the spheres was to have the centers of the spheres in each layer above where the spheres in the layer below kissed.

This is often done with fruit in markets.

This seems so intuitive that it was very difficult to prove mathematically.

Although many tried, including the “prince of mathematics” Johann Carl Friedrich Gauss, it was proved almost four centuries later, in 1998 by the work of Thomas Hales of the University of Michigan and the power of computers.

Even that verification did not convince all mathematicians; Even today there are those who do not find merit in Kepler’s hypothesis.

Those who don’t know the test

And that’s not the only headache caused by spheres.

In fact, a broader category of mathematical problems is called “sphere packing problems”.

Solving them helps in everything from studying the structure of crystals to improving the signals sent by cell phones, space probes and the Internet.

Like Raleigh with its cannonballs, logistics, raw materials and many other industries rely heavily on optimization methods provided by mathematics.

For example, mathematicians discovered that roughly stacked spheres occupy ~64% of any space with density. But if you stack them carefully in certain ways, you can get 74%.

That 10% represents not only transportation costs but also environmental damage.

But practical applications like this require mathematical proof, and the set of spheres, like the Kepler conjecture, throws up particularly difficult unknowns.

One of them stems from a conversation between Isaac Newton, one of the greatest scientists of all time, and David Gregory, the first university professor to teach Newton’s most sophisticated theories.

This is the problem with many kisses, but…

What are they?

Imagine you have several cardboard circles of the same size and need to glue one of them around a board.

The number of kisses equals the maximum number of circles you can place by kissing or touching the center.

It’s that simple.

Well, mathematicians have shown that a maximum of 6 circles can be placed around the beginning, so the number of kisses is 6.

Now imagine that instead of cardboard circles you have rubber balls of the same size.

Question again: What is the maximum number of balls you can place around a center?

Adding that third dimension—volume—makes specifying the number of kisses even more complicated.

And it took two and a half centuries to complicate it.

Newton Y Gregory

The matter began with the famous debate between Newton and Gregory in 1694 on the campus of Cambridge University.

Newton was already 51 years old, and Gregory visited him for several days, during which they talked incessantly about science.

The conversation was one-sided, and Gregory took notes of everything the Grandmaster said.

One of the points discussed and recorded in Gregory’s notebooks is how many planets orbit the Sun.

From there, the discussion went on a tangent to the question of how many spheres of the same size can be arranged in concentric layers touching a center.

Gregory stated – without much preamble – that the first layer around a central ball consisted of a maximum of 13 spheres.

Number of kisses for Newton is 12.

Gregory and Newton never agreed, and neither knew what the right answer was.

Today, the maximum number of spheres a power plant can kiss is often called “Newton’s number” to reveal who is right.

The debate ended in 1953 when German mathematician Kurt Schutte and Dutch BL van der Weerden showed that the number of kisses in three dimensions is 12 and only 12.

The question is important because a group of packed spheres has an average number of kisses, which helps describe the situation mathematically.

But there are unresolved issues.

A thousand kisses

Beyond dimensions 1 (space), 2 (circles), and 3 (spheres), the kissing problem is almost unsolved.

There are only two other instances where the number of kisses is known.

In 2016, Ukrainian mathematician Marina Vyasovska established that the number of kisses in dimension 8 is 240, and in dimension 24 it is 196,560.

As for other dimensions, mathematicians slowly reduce the possibilities to narrower ranges.

For more than 24 dimensions or a general theory, the problem remains open.

There are many obstacles to a complete solution, including computational limitations, but incremental progress on this problem is expected in the coming years.

But what is the use of enclosing spheres of dimension 8, for example?

Algebraic topology specialist Jam Aguday answered that question in a 1991 paper entitled “One Hundred Years of E8”.

“It is used to make phone calls, listen to Mozart on compact disc, send faxes, watch satellite television, and connect to a computer network through a modem.

It is used in all processes that require efficient transfer of digital information.

“Information theory teaches us that signal transmission codes are more reliable in higher dimensions and the E8 lattice, given its surprising symmetry and the existence of an appropriate decoder, is a fundamental tool in code theory.” and signal transmission.

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